Thursday, August 31, 2006

One of the key passages for modern mereology is Geach's creation of the cat on the mat problem in Reference and Generality:

How is "---- is the same A as ----" related to a proper name for an A? To attack this problem, I shall first set forth a paradox that I developed from a sophisma of William of Sherwood.

The fact cat sat on the mat. There was just one cat on the mat. The cat's name was "Tibbles": "Tibbles" is moreover a name for a cat.--This simple story leads us into difficulties if we assume that Tibbles is a normal cat. For a normal cat has at least 1,000 hairs. Like many empirical concepts, the concept (single) hair is fuzzy at the edges ; but it is reasonable to assume that we can identify in Tibbles at least 1,000 of his parts each of which definitely is a single hair. I shall refer to thee hairs as h1, h2, h3...up to h1,000.

Now let c be the largest continuous mass of feline tissue on the mat. Then for any of our 1,000 cat-hairs, say hn, there is a proper part cn of c which contains precisely all of c except the hair hn; and every such part cn differs in a describable way both from any other such part, say, cm, and from c as a whole. Moreover, fuzzy as the concept cat may be, it is clear that not only is c a cat, but also any part cn is a cat: cn would clearly be a cat were the hair hn plucked out, and we cannot reasonably suppose that plucking out a hair generates a cat, so cn must already have been a cat. So, contrary to our story, there was not just one cat called "Tibbles" sitting on the mat; there were at least 1,001 sitting there! Of course this would involve a great deal of overlap and sharing of organs among these 1,001 cats, but logic has nothing to say against that; after all, it happens on a small scale between Siamese twins.

All the same, this result is absurd....

Everything falls into place if we realize that the number of cats on the mat is the number of different cats on the mat; and c13, c279, and c are not three different cats, they are one and the same cat. Though none of these 1,001 lumps of feline tissue is the same lump of feline tissue as another, each is the same cat as any other: each of them, then, is a cat, but there is only one cat on the mat, and our original story stands.

Geach then points out that this appears to commit us to the claim that "---- is the same cat as ----" is an equivalence relation, not an absolute identity relation; which, of course, he has no problem with, since he doesn't think there is any such thing as absolute identity. He also argues that it shows "that logic can and must avoid assuming a syntactical category of proper names." A proper name for an A may be the shared name of several Bs, if Bs are in some sense the same A as any of the others. An interesting argument, although I'm not sure this is a wise argument for this conclusion. I'll have to look more closely at what Geach says about proper names.

In any case, I wonder which of William's sophismata Geach has in mind, and whether he mentions it less vaguely elsewhere.

Tuesday, August 29, 2006

In this world our prudence is rewarded, and our imprudence punished; the one by satisfaction and external advantages, the other by inconveniences and sufferings. These afford instances of a right constitution of nature.

Joseph Butler, Analogy of Religion. (The edition linked to is Hobart's Analysis, which was designed to help students study Butler's Analogy -- which was a major textbook at many schools in the nineteenth century -- for the purposes of passing examinations. Like CliffsNotes, only more thorough and less misleading.)

Reading this post at "Language Log" about the recent fuss over whether Pluto should be called a planet has led me to wonder: why don't scientists do more work with linguists on issues like this? If they had half the sense of Faraday, Maxwell, or any other of the major Victorian scientists, they would. It's common to find an attitude among scientists today that these purely verbal issues aren't important; but, to be quite frank, this is in most cases just because they have been spoiled by having most of the major terminological problems in their fields worked out long ago by people smarter than themselves. One of the refreshing things about Victorian scientists was that they appreciated the importance of words. They knew from their work, and their predecessors' work, in physics, chemistry, and biology, that good choices of words can simplify scientific work, both research and pedagogy, immensely. Bad choices of words are not trivial; they add up, bit by bit, and they slow scientific progress down. The more your words fit what you are trying to convey, the more durable your word choices are, the more recognizable your terminology would be to an educated person, the better things work: students learn faster, the public misunderstands less often, and a few scientists might be surprised at how much easier it is to communicate ideas when the words you have available for communication are less arbitrary and more sensible.

An ideal example is Faraday. When he chose the words "anode" and "cathode", he didn't choose them arbitrarily. He wanted words that would fit, that would convey at once to most educated people their relation to each other, and yet would be distinctive enough not to introduce confusions. He consulted William Whewell, asking him what he thought of the words 'eastode' and 'westode'. Whewell was one of the foremost educators of his day; he was the major historian of science of the nineteenth century; and his Philosophy of the Inductive Sciences has a hefty section closely examining the desiderata for scientific terminology. Whewell suggested 'anode' and 'cathode'. Most educated people would have had at least some Greek and would recognize immediately the opposed root meanings ('path up' and 'path down'), they would be consistent with common scientific naming practices, and they would be distinctive enough not to cause confusion. And 'anode' and 'cathode' it was. (Whewell, of course, had no way of foreseeing a day like ours, in which very few educated persons know any Greek at all, and in which 'anode' and 'cathode' therefore could not pull any weight as terms.)

Monday, August 28, 2006

Today is the Feast of St. Augustine, and although I've suddenly become extremely busy (and probably will be so for the next few days, so expect lighter blogging), I would be remiss if I didn't note the day.

One especially worthwhile link (which I had seen before, but of which I was reminded by this post) is an Orthodox take on Augustine. Augustine, of course, is not theologically popular among the Orthodox at all; and because of a noisy group of Augustine-haters in some Orthodox circles, it's easy to get the impression that it's the standard Orthodox view that Augustine was a heretic, plain and simple. In fact, the matter is more complex, and the websited to which I've linked gives the reasons. Saints, of course, are fallible, and it is a sign of a healthy sense of sainthood to be able to recognize that someone you think was very wrong on some very important issues. Sainthood is not a matter of never saying the wrong thing, nor is it a matter of never explaining things the wrong way; it's a matter of fidelity to the mind of Christ.

I find myself in an interesting position of all this. While I don't quite fit standard labels (crazy people rarely do), someone could reasonably call me a Thomist, which always involves a hefty dose of Augustine, even if in a modified and organized form. I can also, however, be called a Palamist, since I think Palamas's development of the Patristic doctrine of divine energies is reasonable and well-founded. The combination is a rare one, because they are usually considered to be inconsistent with each other, at least on any major point (obviously I don't think so; but I am in a very, very small minority, both East and West).

I find it worth noting that the overwhelming majority of criticisms of Augustine, on whatever position, can easily be shown to rest on misreadings or to be based on assumptions that need not be accepted. He's the saint people love to misrepresent.

For a correction on common views about what Augustine says about sex and sin (and by someone who disagrees with Augustine's views on both), see this essay by David Hunter.

For a correction on common views about Augustine's misogyny (namely, that he was a misogynist), see this essay by Edmund Hill. I always find this calumny completely mystifying. Augustine does have aspects that can be considered sexist in a general way, but he's still one of the least sexist men of his time. The charge of misogyny, which I've occasionally heard, is flabbergasting, given that it's pinned on a man who adored his mother, who explicitly recognizes that many women are better than most men, and more than once argues vehemently for the conclusion that women and men are equal in the eyes of God. One wishes that he had had more influence on Western perceptions of gender than he did.

By no means is Augustine flawless; but he is of overwhelming importance, and no one who reads him fairly and carefully can come away without being the better for it.

At "Cliopatria", Ralph Luker has a list of currently known women history bloggers. (Currently known at the time, that is; a few have been added in the comments, and who knows how many might be starting at any time.) There were quite a few I'd never come across before, which reminds one of just how immense the blogosphere is.

In Part I, I noted the basics of SETL, in a rough way.In Part II and Part III, I discussed briefly some special cases and how SETL handles them.In Part IV, I discussed some basics of argument using SETL.

Up to this point we have largely seen how SETL and modern predicate logic are equivalent (or, if Purdy is right, we have seen how SETL and modern predicate logic are equivalent except for cases of identity between variables; see Part II for my discussion of this). What I'd like to do here, following some of the work by Sommers, is to look at a few very simple sorts of argument that shed light on the differences between SETL and modern predicate logic. It will be seen that SETL has a few advantages, at least minor advantages, that modern predicate logic does not. (As noted before, sources and supplementary readings can be found in a post to come.)

General Points

First, it might be useful to note the obvious differences between the way we handle propositions in predicate logic and the way we do so in SETL. In predicate logic, the following are true:

(1) The subject is always singular.(2) The predicate is always general.(3) Denial of the predicate is treated as being the same as propositional negation.(4) Affirmations are treated as logically simple, while denials are treated as logically complex.(5) Every term is treated as if it were positive.

None of these are constraints on SETL. In SETL, both subjects and predicates can be singular or general; denial of the predicate is distinguished from propositional negation; affirmations and denials can be treated as being at the same logical level; terms may be positive (red) or negative (unred) and this can be portrayed in the logical formulation itself. What this, taken altogether, means is that SETL can take as simple things that predicate logic has to take as secretly complex. For instance:

All dogs are canines.

In SETL this is the simple proposition, -D+C. In predicate logic, we have to translate it as (x)(Dx → Cx). Or to give it in a rough, regimented English version, "For anything: if it is a dog, it is a canine." This is not simple at all; we have two predicates (where SETL has only one), an implication (which SETL doesn't need), and a variable (which SETL also doesn't need). The standard response is that the sentence, "All dogs are canines" has a 'logical form' that is more complex than we might think from the natural language version. And, indeed, this is true if we are putting things into predicate logic. But we don't have to do so; and when we use SETL it doesn't have a secretly complex logical form at all.

Simplification Arguments

Consider the following argument: John smiles at Laura; therefore, John smiles. This is easily handled by SETL (if we formulate a rule of association and a rule of simplification):

1. ±J+(S±L) premise2. (±J+S)±L Assoc from 13. ±J+S Simp from 2

Similarly, for "John smiles at some person; therefore, John smiles":

1. ±J+(S+P) premise2. (±J+S)+P Assoc from 13. ±J+S Simp from 2

Sommers notes, with regard to arguments of this last type, that they baffle attempts to formulate them in modern predicate logic. You can represent (1) as (∃x)(Sjx); but there seems to be no way to represent (3) in a different form. Of course, this is not a problem for modern predicate logic, since one could just say that this shows why the inference is a good one -- the 'logical form' of (1) and (3) are the same. But when we have a singular direct object, as with the first, it isn't quite clear -- although probably still could be argued -- that modern predicate logic handles it properly. The natural way to handle (1) would be to translate it as (Sjl); we can then get the conclusion, (3) by existential generalization, as (∃x)(Sjx). But this means that the first argument above is a different argument entirely from the second, and one can argue (as Sommers does) that it would make sense if the two arguments above were treated exactly the same way. The reason modern predicate logic fails to do this is that it, unlike SETL, lacks a predicative functor that joins the subject and predicate; it assumes predication as a given.

Someone Thomas studied was the disciple of Aristotle.Therefore, Thomas studied the disciple of Aristotle.

These are easily handled in SETL by association. The first argument becomes:

1. (±T1+S12)±A2premise2. ±T1+(S12±A2) Assoc from 1

And the second:

1. (±T1+S12)+(D±A2) premise2. ±T1+(S12+(D±A2)) Assoc from 1

The form of argument is very simple and straightforward in both cases. To handle either of these arguments, however, modern predicate logic has to appeal to identity and Leibniz's Law. So, for instance, it would handle the first argument in the following way:

1. (∃x)(Rwx & x=a) premise2. Rwa Leibniz's Law from 1

With a slightly more complicated premise, you can do the same with the second argument. There is no doubt that it can handle this form of argument; but it has to do so in a roundabout way, whereas SETL handles it very straightforwardly.

In predicate logic, however, we can't represent the argument at all unless we treat the premise has having the same logical form as the conclusion. Thus we are in the same situation for passive transformation as we were for simplification. This is not surprising; Frege, for instance, explicitly considers passive transformation in order to conclude that it is not significant for logical matters because the active and passive forms have the same truth conditions. Sommers replies to this that then we should regard the change from "Some A is B and C" to "Some C is B and A" as insignificant for logical purposes, because they have the same truth conditions; and that the more powerful concept-script is the one that brings the most transformations within the field of what is significant for logical purposes. And as we've done with active and passive voice, so can we do with dative movement (e.g., "Dave sold the truck to June; therefore Dave sold June the truck").

None of this is particularly fearsome for the die-hard fan of modern predicate logic. Indeed, the reason that SETL handles these arguments so easily is that it is much closer to natural language than predicate logic is, and, of course, they are quite ordinary arguments in natural language. But it should give one some pause, at least to this extent: there are common arguments that can be expressed easily in SETL (if we decide that they are important enough to be expressed in it) that are not obviously handled properly in predicate logic.

So that's a brief survey of some of the differences. The next post on this subject will look at how one might extend basic SETL into modality.

In Part I, I gave a crude characterization of the basics of SETL.In Part II, we saw a few refinements.In Part III, we looked at a few loose ends.Now we get to the good stuff: actual inferences.

Tautology

Following Aristotle, we will call a perfect inference any inference that does not require a proof of its validity (because it is clearly valid); one in need of a proof of validity is imperfect. All basic rules of inference are patterns for perfect inferences. For our purposes we will not count as perfect inferences anything with more than two premises, however obviously valid it may be. Thus there are three types of perfect inference: two-premise perfect inferences, one-premise perfect inferences, and zero-premise perfect inferences. It may seem odd to talk about a 'zero-premise' inference; but what is meant is that you don't need a premise to introduce it. Thus all tautologies are zero-premise inferences; they can be introduced as you please.

We know that there are two necessarily contradictory types of proposition:

All S aren't S.Some S aren't S.

Denying these, we get:

All S are S.Some S are S.

These are tautologies; as will be "All S that are M are S" and "Some S that are M are S". From this we can formulate our first rule of inference, the Rule of Identity (Id):

∴ ±(±S±M)+ (±S)

where M may be empty (i.e., discardable). (In this rule, as in all the others we will use, ± means 'either plus or minus', and the assumption is that you will be consistent in which you choose.) Basically, the rule is that we may introduce a universal or particular tautology whenever we wish, without regard for prior premises. This will be the only zero-premise perfect inference we require.

Immediate Inference

A single-premise perfect inference is also called an immediate inference. The most obvious rule for such inferences is the Rule of Self Inference (SI):

±S±P ∴ ±S±P

The second rule of immediate inference has to do with universal affirmations; it is the Rule of Contraposition (Contrap), which I will break into two parts purely for the sake of clarity:

-(+S)+(+P) ∴ -(-P)+(-S)-(-S)+(-P) ∴ -(+P)+(+S)

Basically this says that you can infer "All nonP is nonS" from "All S is P", and vice versa.

The third rule of inference has to do with particular affirmations; it is the Rule of Conversion (Con):

+(±S)+(±P) ∴ +(±P)+(±S)

So we have rules governing affirmations. But we need rules for doing things with denials. The easiest way to do this is to formulate a rule for converting denials into affirmations; then the rules we have already will apply for all our needs. The Aristotelian claim that any denial is equivalent to an affirmation is very plausible. So we can formulate a Rule of Equivalence (Eq), which again I will split into two parts for clarity:

±S-(+P) ∴ ±S+(-P)±S-(-P) ∴ ±S+(+P)

Related to this, we have a Rule of Double Negation (DN) to simplify and expand expressions; any (T) term can expand to (-(-T)) and any (-(-T)) can simplify to (T).

A useful additional rule is the Rule of Commutation (Com), which basically says that for any terms we can rewrite (P+Q) as (Q+P). If we want to, we can throw in a Rule of Association (Assoc) to handle groupings, so that S+(M+P), for instance, would be equivalent to (S+M)+P. There is actually some reason to do this explicitly(we can handle more types of inferences if we do, as we will see in a later post), but I won't bother much about the rule, since parentheses are largely a matter of convenience in translating. It's also possible to have a Rule of Simplification (Simp), to handle inferences like "John smiles at Jane; therefore John smiles." We'll use it later, but I won't develop this issue here, either.

One last, and very useful, rule of immediate inference is the Rule of Composition (Comp):

-S+P ∴ -(S+Q)+(P+Q)

where Q is an adjective term (whether monadic or relational doesn't matter). This allows you to handle inferences like "All circles are figures; therefore all drawers of circles are drawers of figures" and "All men are mortal; therefore all rational men are rational mortals."

Mediate Inferences

With immediate inferences in hand we are ready to tackle syllogistic, which in term logic covers all mediate inferences (two premises or more). To do this we need to add just one basic rule, the extremely important dictum de omni et nullo (DDO), which can be put in the same form as our other rules to constitute a Rule of Mediate Inference (MI):

-M+P, ±S+M ∴ ±S+P

It may not be obvious what's going on here. The basic point is this: whatever is affirmed of all of something is likewise affirmed of what that something is affirmed of.

Given this, we are doing swell. We can show that the syllogism Darapti is valid by the following proof:

Thus every Darapti syllogism, i.e., every mediate inference of the form "All M is P; All M is S; therefore some S is P" is valid.

Using propositional nominalization and meta-propositional predicate we can prove that modus tollens is valid:

1. -[p]+[q] premise2. -[q]+(-E) premise3. -[p]+(-E) MI from 1,2

And likewise with modus ponens:

1. -[p]+[q] premise2. +[p]+(E) premise3. +[q]+(E) MI from 1,2

It's easy enough to identify the necessary and sufficient conditions for validity in SETL. They are:

(1) A universal conclusion is only validly drawn from universal premises.(2) A particular conclusion is only validly drawn from a set of premises that contains one and only one particular premise.(3) The premises in a valid inference sum up to the conclusion.

So this is invalid:

-M-(-S)-(-P)+S∴ -M+P

(2) is irrelevant; it meets (1), but the sum of the premises is -M+P+S. This is also invalid:

+S+(-M)+M+P∴ +S+P

(1) is irrelevant; it meets (3), but it tries to draw a particular conclusion from two particular conclusions.

Relational Arguments

So far, so good. It might not be immediately obvious how one would handle relational premises, though. Consider the following inference:

Some atheist mocks every prayer.Every Shiite recites a prayer.∴ Every Shiite recites what some atheist mocks.

In SETL the premises will be:

1. +A1+(M12-P2) 2. -S3+(R32+P2)

The middle term here is P2. Thus (pretty much ignoring the parentheses, which I only put in here to make it easier to translate without accidental shift of meaning) by MI we can conclude:

3. +A1+(M12-S3+R32)

This is clearly a valid syllogism; the first condition for validity is irrelevant, and it meets both the second and the third. And taking this conclusion, by Com we get:

4. -S3+(R32+A1+M12)

So that's a rough, quick look at inferences in SETL. In the next post I will look briefly at a few simple things that you can do (if you choose) in SETL more easly than in ordinary predicate logic, in order to show that SETL is better able to capture natural-language inferences (if that's our goal).

Caveats

For a rough introduction to my philosophy of blogging, including the Code of Amiability I try to follow on this weblog, please read my fifth anniversary post. I consider blogging to be a very informal type of publishing - like putting up thoughts on your door with a note asking for comments. Nothing in this weblog is done rigorously: it's a forum to let my mind be unruly, a place for jottings and first impressions. Because I consider posts here to be 'literary seedings' rather than finished products, nothing here should be taken as if it were anything more than an attempt to rough out some basic thoughts on various issues. Learning to look at any topic philosophically requires, I think, jumping right in, even knowing that you might be making a fool of yourelf; so that's what I do. My primary interest in most topics is the flow and structure of reasoning they involve rather than their actual conclusions, so most of my posts are about that. If, however, you find me making a clear factual error, let me know; blogging is a great way to get rid of misconceptions.